It has been observed that racks and Leibniz algebras come with a new kind of homology. These invariants were first defined intrinsically, and then it was realized that they have a natural interpretation in terms of the corresponding cubical and DG structures. This paper aims to point out that there is a similar homology for pre-crossed modules. Since both pre-crossed modules and augmented racks are one-dimensional shadows of associative products, the former being simplicial while the latter pre-cubical, it is natural that these new invariants are related to the rack homology.
An augmented rack \[ \pi:X\rightarrow G \] gives rise to the pre-crossed module \[ \overline{\pi}:F\left( X\right) \rightarrow G \] where \(F\left( X\right) \)is the free group on \(X\)and \(\overline{\pi}\)is induced by \(\pi\). The principal result in this paper is Theorem 3.1 claiming that the pre-crossed homology of this pre-crossed module coincides with the rack homology \(X\overset{\pi}{\rightarrow}G\). Another case when one can identify the pre-crossed homology is that of a pre-crossed module \(X\overset{\pi}{\rightarrow}G\)with a trivial action, where it turns out to be the tensor algebra on the homology of the group \(X\) (Theorem 4.1).
In order to relate the rack homology to the pre-crossed homology the author uses the group-completion theorem of Quillen in the appendix to [E. M. Friedlander and B. Mazur, Filtrations on the homology of algebraic varieties. Providence, RI: American Mathematical Society (AMS) (1994; Zbl 0841.14019)]. As for the computation of the homology for the pre-crossed modules with the trivial action, it rests on the identification of the classifying space for the pre-crossed homology as a certain twisted version of the Milnor-Carlsson construction of a circle [G. Carlsson, Topology 23, 85–89 (1984; Zbl 0532.55024)].